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21
Facts and Challenges from the Great Recession for Forecasting and Macroeconomic Modeling
, 2013
"... This paper provides a survey of business cycle facts, updated to take account of recent data. Emphasis is given to the Great Recession which was unlike most other postwar recessions in the US in being driven by deleveraging and financial market factors. We document how recessions with financial mar ..."
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This paper provides a survey of business cycle facts, updated to take account of recent data. Emphasis is given to the Great Recession which was unlike most other postwar recessions in the US in being driven by deleveraging and financial market factors. We document how recessions with financial market origins are different from those driven by supply or monetary policy shocks. This helps explain why economic models and predictors that work well at some times do poorly at other times. We discuss challenges for forecasters and empirical researchers in light of the updated business cycle facts.
Forecasting mixed frequency time series with ecmmidas models
, 2012
"... This paper proposes a mixedfrequency errorcorrection model in order to develop a regression approach for nonstationary variables sampled at different frequencies that are possibly cointegrated. We show that, at the model representation level, the choice of the timing between the lowfrequency dep ..."
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This paper proposes a mixedfrequency errorcorrection model in order to develop a regression approach for nonstationary variables sampled at different frequencies that are possibly cointegrated. We show that, at the model representation level, the choice of the timing between the lowfrequency dependent and the highfrequency explanatory variables to be included in the longrun has an impact on the remaining dynamics and on the forecasting properties. Then, we compare in a set of Monte Carlo experiments the forecasting performances of the lowfrequency aggregated model and several mixedfrequency regressions. In particular, we look at both the unrestricted mixedfrequency model and at a more parsimonious MIDAS regression. Whilst the existing literature has only investigated the potential improvements of the MIDAS framework for stationary time series, our study emphasizes the need to include the relevant cointegrating vectors in the nonstationary case. Furthermore, it is illustrated that the exact timing of the longrun relationship does not matter as long as the shortrun dynamics are adapted according to the composition of the disequilibrium error. Finally, the unrestricted model is shown to suffer from parameter proliferation for small sample sizes whereas MIDAS forecasts are robust to overparameterization. Hence, the datadriven, lowdimensional and flexible weighting structure makes MIDAS a robust and parsimonious method to follow when the true underlying DGP is unknown while still exploiting information present in the highfrequency. An empirical application illustrates the theoretical and the Monte Carlo results.
Nowcasting and the realtime data flow �
, 1564
"... In 2013 all ECB publications feature a motif taken from the €5 banknote. NOTE: This Working Paper should not be reported as representing the views of the European Central Bank (ECB). The views expressed are those of the authors and do not necessarily reflect those of the ECB. Acknowledgements The au ..."
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Cited by 4 (0 self)
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In 2013 all ECB publications feature a motif taken from the €5 banknote. NOTE: This Working Paper should not be reported as representing the views of the European Central Bank (ECB). The views expressed are those of the authors and do not necessarily reflect those of the ECB. Acknowledgements The authors would like to thank G. Elliott, A. Timmermann and two anonymous referees for their comments. This research is supported
A survey of econometric methods for mixedfrequency data.
, 2013
"... Abstract The development of models for variables sampled at di¤erent frequencies has attracted substantial interest in the recent econometric literature. In this paper we provide an overview of the most common techniques, including bridge equations, MIxed DAta Sampling (MIDAS) models, mixed frequen ..."
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Abstract The development of models for variables sampled at di¤erent frequencies has attracted substantial interest in the recent econometric literature. In this paper we provide an overview of the most common techniques, including bridge equations, MIxed DAta Sampling (MIDAS) models, mixed frequency VARs, and mixed frequency factor models. We also consider alternative techniques for handling the ragged edge of the data, due to asynchronous publication. Finally, we survey the main empirical applications based on alternative mixed frequency models. J.E.L. Classi…cation: E37, C53 Keywords: mixedfrequency data, mixedfrequency VAR, MIDAS, nowcasting, forecasting We would like to thank Tommaso Di Fonzo, Eric Ghysels, Helmut Lutkepohl for useful comments on a previous version. The views expressed herein are solely those of the authors and do not necessarily re ‡ect the views of the Norges Bank. The usual disclaimers apply.
2014), “Do HighFrequency Financial Data Help Forecast Oil Prices? The MIDAS Touch at Work,” forthcoming
 International Journal of Forecasting
"... In recent years there has been increased interest in the link between financial markets and oil markets, including the question of whether financial market information helps forecast the real price of oil in physical markets. An obvious advantage of financial data in forecasting monthly oil prices i ..."
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In recent years there has been increased interest in the link between financial markets and oil markets, including the question of whether financial market information helps forecast the real price of oil in physical markets. An obvious advantage of financial data in forecasting monthly oil prices is their availability in real time on a daily or weekly basis. We investigate the predictive content of these data using mixedfrequency models. We show that, among a range of alternative highfrequency predictors, cumulative changes in U.S. crude oil inventories in particular produce substantial and statistically significant realtime improvements in forecast accuracy. The preferred MIDAS model reduces the MSPE by as much as 28 percent compared with the nochange forecast and has statistically significant directional accuracy as high as 73 percent. This MIDAS forecast also is more accurate than a mixedfrequency realtime VAR forecast, but not systematically more accurate than the corresponding forecast based on monthly inventories. We conclude that typically not much is lost by ignoring highfrequency financial data in forecasting the monthly real price of oil.
A MultiCountry Approach to Forecasting Output Growth Using PMIs*
, 2014
"... This paper derives new theoretical results for forecasting with Global VAR (GVAR) models. It is shown that the presence of a strong unobserved common factor can lead to an undetermined GVAR model. To solve this problem, we propose augmenting the GVAR with additional proxy equations for the strong fa ..."
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This paper derives new theoretical results for forecasting with Global VAR (GVAR) models. It is shown that the presence of a strong unobserved common factor can lead to an undetermined GVAR model. To solve this problem, we propose augmenting the GVAR with additional proxy equations for the strong factors and establish conditions under which forecasts from the augmented GVAR model (AugGVAR) uniformly converge in probability (as the panel dimensions N,T → ∞ such that N/T→κ for some 0<κ<∞) to the infeasible optimal forecasts obtained from a factoraugmented highdimensional VAR model. The small sample properties of the proposed solution are investigated by Monte Carlo experiments as well as empirically. In the empirical part, we investigate the value of the information content of Purchasing Managers Indices (PMIs) for forecasting global (48 countries) growth, and compare forecasts from AugGVAR models with a number of datarich forecasting methods, including Lasso, Ridge, partial least squares and factorbased methods. It is found that (a) regardless of the forecasting methods considered, PMIs are useful for nowcasting, but their value added diminishes quite rapidly with the forecast
RealTime Forecast Density Combinations∗ Forecasting US GDP Growth Using MixedFrequency Data
, 2012
"... We combine the issues of dealing with variables sampled at mixed frequencies and the use of realtime data. In particular, the repeated observations forecasting (ROF) analysis of Stark and Croushore (2002) is extended to an autoregressive distributed lag setting in which the regressors may be sample ..."
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We combine the issues of dealing with variables sampled at mixed frequencies and the use of realtime data. In particular, the repeated observations forecasting (ROF) analysis of Stark and Croushore (2002) is extended to an autoregressive distributed lag setting in which the regressors may be sampled at higher frequencies than the regressand. For the US GDP quarterly growth rate, we compare the forecasting performances of an AR model with several mixedfrequency models among which the MIDAS approach. The additional dimension provided by different vintages allows us to compute several forecasts for a given calendar date and use them to construct forecast densities. Scoring rules are employed to test for their equality and to construct combinations of them. Given the change of the implied weights over time, we propose timevarying ROFbased weights using vintage data which present an alternative to traditional weighting schemes.
MIxed DAta Sampling) described in Ghysels et
"... Abstract Economic data are collected at various frequencies but econometric estimation typically uses the coarsest frequency. This paper develops a Gibbs sampler for estimating VAR models with mixed and irregularly sampled data. The Gibbs sampler allows efficient likelihood inference and uses simpl ..."
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Abstract Economic data are collected at various frequencies but econometric estimation typically uses the coarsest frequency. This paper develops a Gibbs sampler for estimating VAR models with mixed and irregularly sampled data. The Gibbs sampler allows efficient likelihood inference and uses simple conjugate posteriors even in high dimensional parameter spaces, avoiding a nonGaussian likelihood surface even when the Kalman filter applies. Two examples studying the relationship between financial data and the real economy illustrate the methodology and demonstrates efficiency gains from the mixed frequency estimator. Keywords: Gibbs Sampling, Mixed Frequency Data, VAR, Bayesian Estimation JEL classification: C11, C22, G10, E27 2 Economic data are rarely collected at the same instances in time. Data from liquid markets are available almost continuously, while aggregate macro data in many cases are available only at monthly, quarterly, or annual frequencies. Mixed and irregular sampling frequencies represent a significant challenge to timeseries econometricians. This paper develops Bayesian estimation of mixed frequency Vector Autoregressions (VAR's). The method is a simple, yet very powerful algorithm for MarkovChainMonteCarlo sampling from the posterior distributions of the VAR parameters. The algorithm works in the presence of mixed frequency or irregularly spaced observations. The posterior is conditioned on data observed at mixed frequencies rather than simply data observed at the coarsest frequency. The method follows from the assumption that the econometrician simply does not observe the high frequency realizations of the low frequency data, and can accordingly treat these data as missing values. Consequently, and consistent with the standard utilization of missing values in Bayesian econometrics, the Bayesian Mixed Frequency (BMF) algorithm developed is a Gibbs sampler that produces alternate draws from the missing data and the unknown parameters in the model. Under typical assumptions about normally distributed exogenous shocks, the VAR's linear structure allows for draws from Gaussian conditional distributions for estimating the missing data, along with draws from Gaussian and inverse Wishart conditional posterior distributions for the parameters in the model. Since this Gibbs sampler requires only simulation from known densities, it is extremely simple to implement. There has been much work addressing the issue of mixed frequency data from a variety of different approaches. An early contribution is the Kalman filtering approach introduced by Harvey and Pierse (1984), which notes that for linear VAR models, missing observations can be incorporated by simply skipping a term from the updating equation whenever an observation is missing. The VAR's linear and Gaussian form makes it straightforward to formulate a statespace form. However, the Kalman filter approach is potentially cumbersome when the missing data occur at irregular frequencies, especially if there are multiple 3 series with missing data at differing frequencies. In addition, the Kalman filter yields a likelihood function that is nonlinear and nonGaussian over a potentially very large parameter space; analyzing such likelihood functions often proves difficult both from frequentist and Bayesian viewpoints. The BMF approach, by contrast, handles irregular and multiple missing series with ease, and the Gibbs sampling from standard densities makes the analysis of the resulting posterior densities very tractable. Another approach, suggested by A growing body of work considering the estimation of mixed frequency models is the work on MIDAS (MIxed DAta Sampling) described in Ghysels et al. substantially from the Kalman filter approach of Harvey and Pierse (1984), it potentially suffers from the same pitfalls: handling observations that are irregularly spaced requires altering the estimated equations as in In contrast to these methods, the approach taken in this paper is from a Bayesian perspective, and consequently treats lower frequency data as missing. The missing data approach to higher frequency data has a history from both a Bayesian and frequentist perspective. Chow and Lin (1971) discuss how to interpolate time series using related series. The traditional approach for dealing with mixed frequency data is to discard high frequency data and simply perform estimation on the coarsest frequency data. This estimation strategy potentially discards information contained in the higher frequency data, yet is used often in macro time series econometrics, especially within the context of VAR estimation, making it a useful benchmark. Indeed, a number of Bayesian and frequentist applications, including studying the effects of monetary policy, oil, or uncertainty shocks, include VAR's estimated at a monthly frequency despite the availability of higherfrequency data. The coarse estimation can be used to identify parameters in the VAR even if the econometrician assumes that the true VAR evolves at some higher frequency than that used for estimation because Gaussian VAR's are closed under temporal aggregation. In addition to developing the methodology, this paper demonstrates the advantages of the BMF estimation method using numerical simulations and actual data. For numerical simulations over a range of parameter constellations, BMF dominates estimation using coarse 5 sampling from the frequentist perspective of absolute bias and mean squared deviations from the true values. The simulations suggest that BMF is particularly beneficial when the series have correlated shocks and when the sample size is small. After considering simulated data, the paper studies the relationship between financial markets and the real economy uncovered by BMF and by the coarser sample estimator. Two examples, one using monthly financial data with annual GDP and one using weekly financial data with monthly industrial production, show that BMF has quantitative and qualitatively different implications from those of the coarser sample estimator. In this context, BMF outperforms the coarsely sampled estimator in that the posterior standard deviations are smaller when using BMF. The BMF approach also improves the estimation of impulse response functions, as the decrease in parameter uncertainty associated with BMF is typically reflected in tighter confidence bands for the impulse response functions. Among other things, this result implies that BMF can allow for sharper conclusions about the impact of economic policies or the effects of shocks. The remainder of the paper is organized as follows: Section 1 discusses the construction of a Gibbs sampler for the model. Section 2 presents simulation based evidence for the advantages of using the BMF approach. Sections 3 and 4 show empirical applications that study the effects of financial markets on the real economy. Finally, Section 5 concludes. Econometric Methodology This section discusses the main algorithm of data augmentation and estimation in the presence of missing data. The model is where dim (y t ) = N . Denote the set of parameters Θ = (A, B 1 , . . . , B k , Σ), where dim (x t ) = N x and dim (z t ) = N z such that N z + N x = N and suppose x t is a fully 6 observed variable and z t is a variable with missing data. For simplicity, focus on the case when k = 1 and assume that z and x are recorded at two frequencies, but note that the method applies to a multifrequency dataset. In the simulated data application in Section 2, x t is observed monthly and z t is observed quarterly; the example in Section 3 has x t observed monthly and z t observed annually; the example in Section 4 has x t observed weekly and z t observed monthly. In the case of monthly and quarterly observations, the missing data are {ẑ 1 ,ẑ 2 ,ẑ 4 ,ẑ 5 ,ẑ 7 , ...}, whereẑ t denotes a sampled observation at time t. Letẑ denote the vector of observed and sampled data,ẑ \t denote all elements ofẑ except the tth ones, and letŶ (i) denote the full collection of observed and sampled data at iteration i. The BMF estimator is an application of Bayesian Gibbs sampling, which requires iterating over objects of interest, sampling those objects from known distributions conditional upon the remaining objects. In the current setup, the objects of interest are the missing observations, the matrices A and B 1 , and the covariance matrix Σ. Given prior distributions and initial values of the parameters, the ith iteration of the MCMC algorithm reads • Step 1 : for t = 1, .., T , draw missing dataẑ is the vector of most recently updated missing values and are the latest draws of A, B 1 , Σ, respectively. For example, in the case of consecutive updating,ẑ The new step in the procedure is Step 1, which is drawing missing data given the parameters in the model and the fully observed data. Except for this first step, the procedure is a standard Normal linear model which, under conjugate priors, yield Normal and inverse Wishart posterior distributions. Since drawing from the relevant posteriors in Steps 2 and 3 7 is a wellknown procedure, the following focuses on Step 1, sampling the missing data given a set of parameters. Step 1: Sampling the Latent Data Step 1 of the Gibbs sampler requires drawing the latent data from its conditional posterior distribution. It is convenient to draw a single tth element in one operation, so the goal is to drawẑ . Appendix A shows that the conditional density for z t is the multivariate normalẑ It is now straightforward to construct Gibbs sampling to drawẑ t , since it is also conditionally normal. One possibility is to draw the elements in a consecutive order. Another approach is to draw odd and even elements of z alternately, which can easily be implemented in a vectorized programming environment. Coarse Sampling Estimation The standard approach to mixed frequency estimation is to delete the high frequency data so that the VAR is estimated at whichever frequency is jointly available. Thus, in estimating a model with, in the case of the example in Section 3, monthly and annual data, one would sample both variables at the annual frequency. In choosing a annual sampling frequency for the monthly data, one throws away information contained in the higher frequency data. It should be noted that, in the context of many macroeconomic applications that use mixed frequency data, many simply perform estimation at the lowest frequency. study the effects of uncertainty shocks, even though some of the relevant asset pricing data are available at much higher frequencies. In each of these applications, discarding data at high frequencies and estimating using the lowest sampling interval is standard procedure. The estimator based solely on coarse data is not an unreasonable estimator. In particular, the estimator can be used to estimate the true values of the parameters in a VAR even if the true VAR evolves at a higher frequency than that used for estimation. This fact follows that Y t+n =Ã n +B n Y t + tn where the new coefficients A n and B n are given bỹ and the covariance matrix of the error tn is Estimation of the lower frequency VAR produces an estimate ofB n , denotedB n , which can then produce an estimate B 1 by computingB 1/n n . Then using Simulation Results Having presented the methodology, this section examines BMF using simulated data. The purpose is to analyze how BMF fares relative to estimation at the coarsest frequency when the objective is to recover parameter estimates, say the posterior mean, that are as close as possible in some sense to the truth. This is very much a frequentist way of thinking, and so the exercise should accordingly be interpreted as a small sample study of the posterior mean as a frequentist parameter estimate. The data are generated by a monthly bivariate VAR, where the variable x is observed every period, and the variable z is observed every third period t = 1, 4, 7, ..., 3T , so T is the number of quarters in the sample, meaning there are 3T months. In the shorter sample, there are T = 20 quarters, and in the longer sample there are T = 80. As Hence, this section shows that BMF has significant gains compared to the usual estimation strategy of using the coarsest frequency data in terms of root mean squared errors and absolute bias. Having provided a comparison on simulated data, the following two Sections illustrate how higher frequency financial market leading indicators can improve inference about the real economy. Empirical Application I: Monthly and Annual Data To demonstrate the advantages of the BMF estimator on real data, this section considers a new dataset consisting of monthly financial market data and annual GDP. These sampling frequencies are relatively coarse, which will help give a parsimonious exposition and also allow a longer time series to study the relationship between real activity and financial market leading indicators. 11 Data The dataset contains annual GDP data from 19252013. The use of annual data circumvents the issues relating to seasonal variations in GDP, and allows a sampling period that covers the Great Depression. To study the effects of financial market data on the real economy, the dataset also includes financial data collected at monthly frequencies. The first financial series is the real rate, defined as a 90 day Treasury yield in excess of the 12 month inflation rate as measured by the CPI. This measure of the short term borrowing rate is highly correlated with the Fed Funds rate. The short term rate is typically procyclical, as for example, a Taylor rule would prescribe that the Federal Reserve would increase the nominal rate in response to an overheating economy and reduce it during recessions, respectively. The second financial series is the slope of the yield curve, defined as the difference between 90 day Treasury yields and 25 year Treasury yields. The data were constructed directly from the CRPS Treasury master file. In the few instances that the longest maturity in the sample is less than 25 years, or longer than 90 days, the longest and shortest maturities available are used, respectively. The slope of the yield curve is of interest because it is commonly believed to predict growth: a negative slope seems to have preceded many recessionary periods, including the Great Depression and the 20082010 recession. The third financial market variable used is the default spread, measured as the difference in Baa and Aaa rated corporate bonds from Moody's. This data set is available from several sources including the Federal Reserves's H15 interest rate survey. The default spread is of interest because it reflects the borrowing costs of risky corporations relative to less risky ones: it is an exante risk premium. In asset pricing models with stochastic volatility, the credit spread is a monotonic (often linear) function of conditional variance (e.g., Duffie and Singleton 12 The value of the put naturally depends onetoone on volatility. In a stochastic volatility world, therefore, timevariation in volatility will drive the price of this put option, which again means that some measure of the aggregate credit spread will be a monotonic function of economywide systematic volatility. The default spread is a natural candidate as a leading indicator because an increase (decrease) in firm borrowing costs would have a direct negative (positive) effect on investments. Default spreads and GDP growth should have a lagged relationship, so if firms' borrowing costs are high today, investments will be smaller and their subsequent output will be lower in the future. Second, while the estimated coefficients relating to the movements between the financial variables are similar across the two estimators, their impact on GDP growth are very dissimilar. For example, the bottom row of the coefficient matrix, which measures the impact of last period's shocks on current GDP growth, shows that the impact of shocks to interest rate level and default spread is particularly strong on subsequent GDP growth using the BMF estimates. In particular, the impact of shocks to the default spread is very negative using BMF, while positive and insignificant using the coarse sampling estimator. Note the very surprising drop in the coefficient of "marginal GDP autocorrelation" (B 4,4 ). For the 13 mixed frequency estimation this coefficient is actually negative while large and more than seven standard deviations way from zero for the annual estimator. This difference implies that after conditioning on the high frequency movements in the financial data, there is no leftover GDP growth rate autocorrelation. The serial dependence in GDP growth is entirely subsumed by high frequency movements in persistent financial variables. among many others, the shock is identified recursively, with the assumption that financial variables respond before real variables. Results 5 As seen, the overall impressions from examining the VAR coefficients are reflected in the impulse response functions. In particular, a positive shock to the real rate results in future negative GDP growth. A shock to the slope of the yield curve leads to negative growth also, but this result is only statistically significant for the short term using the BMF estimator. The default spread also negatively predicts GDP growth and again this result is statistically significant only when using the mixed frequency estimator. From The empirical investigation focuses on the idea that positive (negative) shocks to conditional market risk premia are associated with negative (positive) growth; other empirical measures of conditional risk premia provided inconclusive results. The first candidate was a measure of default free, long term borrowing costs, measured by the difference between a 14 τ maturity (here 20 years) yieldtomaturity, y t,τ and the corresponding expected short rate over the life of the treasury, 1 τ t+τ t E t r s ds. This measure did not provide any significant forecasting power. A second alternative was whether realized stock market volatility and logvolatility, computed using daily S&P 500 returns from 1926present, could predict GDP. The results were inconclusive and depend on the particular volatility measure. There were some indications that stock market volatility and the default spread are substitutes in explaining GDP growth, presumably caused by collinearity between the stock market volatility and the default spread, which correlation is about 0.6. Application II: Weekly and Monthly Data The application in the previous section showed how to use BMF to combine monthly and annual data, this application turns to using weekly financial data to inform monthly industrial production. Data Industrial production is available at a monthly frequency, and measures output in a set of subsectors in the economy. Since these production sectors may be especially influenced by changes in interest rates or oil prices, the high frequency data are measures of the level and slope of the yield curve, as well as spot oil prices. Interest rate conditions may affect production decisions, and oil is often an essential input to production, so it is natural to consider these variables along with IP. A number of papers such as Kilian and Park (2009) or Kilian et al. (2009), to name a few, estimate monthly VARs in order to study the affects of oil shocks, even though the spot oil price changes much more frequently. So the analysis uses the spot real price of West Texas Intermediate crude oil, and a measure of the intercept and slope of the yield curve of interest rates. The slope of the yield curve is defined as the difference in yields between the sevenand oneyear zero 15 coupons. The slope gives expectations about future interest rates and because it has been frequently noted that an inverted yield curve (negative slope) tend to precede recessions. All constant maturity zerocoupon yield data are from the dataset by Gurkaynak et al. (2007). While these variables are available at extremely high frequencies, the analysis below focuses on weekly data. The data run from the first week of Jan1986 to the last week of Jul2011, for a total of 1336 weekly observations and 307 monthly observations. In addition to being able to address the impact of interest rates and oil on industrial production, the choice of weekly intervals presents an interesting challenge for mixedfrequency data. The assumption of timing is the following: the last business day of each week (usually Friday but occasionally Thursday), the yield curve and oil spot prices are observed, and the last Friday (or Thursday) of each month, the twelvemonth growth rate of industrial production is observed. The challenge is that most months will have four weekly observations per month, but there will be some months that have five weeks associated with them. While BMF can handle this irregularly observed data with ease, using a method such as the Kalman filter or MIDAS would require either ignoring the fifth week in these months or changing the structure of the estimated equations in these months. Since the base period considered is a week, the analysis below converts the monthly estimates to their weekly counterparts following the method described in Section 1.2. Results Table 3 displays the estimates for the BMF estimator using weekly observations versus the discarding all but the last week of the month and therefore using only monthly observations. As with the previous application, most of the posterior means are similar across methodologies, and BMF tends to have smaller posterior standard deviations. The notable exception to the similar posterior means are the estimates associated with oil, the third variable in the VAR. Here the biggest reduction is associated with the finely observed oil price. When oil is included at a high frequency, the inclusion adds a lot more information about the dynamics 16 of the VAR at a weekly interval. After noting the gains in accuracy from the parameter estimates, especially for the oil variable, Concluding Remarks This paper considers estimation of VAR's using data sampled at mixed frequencies. The methodology uses Gibbs sampling the unobserved data at the high frequency to generate estimates with generally smaller standard errors. The simulation experiments demonstrate that BMF produces more accurate estimates of model parameters than the basic approach of subsampling at the coarse data frequency, and the example application shows that using higher frequency data may produce sizable gains. Improved accuracy is not the only advantage of the BMF estimator. Another benefit is the ability to update forecasts of a coarsely observed variable in response to new arrival of data measured at high frequencies. Along the lines of the application presented above, examples include updating forecasts of GDP in response to monthly measurements of data 17 or using weekly or even daily financial data to forecast aspects of the real economy. The BMF framework allows for a natural approach to incorporate high frequency observations to the low frequency forecasts, which would avoid the use of adhoc forecast revisions. One potential advantage of the Bayesian simulation approach is that it easily generalizes to more complicated models. and the observation equation changes from if z t is observed, to if z t is not observed. In the case of mixed frequency data where the missing data occur at regular frequencies, such as the monthly and quarterly application, the observation equation switches between (8) and BMF then proceeds to estimate this model by simulating, as before, the sparsely observed elements of Y, but in addition treats X as an unobserved variable a variable observed with zero frequency. Importantly, the algorithm for drawing the missing data applies directly in this setting. To proceed to the second step of the Gibbs sampler which involves drawing the parameters, the algorithm needs only slight modifications to impose the zeroconstraints on B * . Note that the estimation of VARMA models can be implemented using this approach. This paper has also not considered the outofsample forecasting ability of BMF estimators. Of course, given the applications, forecasting is a natural extension given mixed frequency data. As with any VARbased method, forecasting given BMF estimates involves iterated forecasting rather than direct forecasting. Finally, while the BMF algorithm applies in general, identification considerations must, as usual, be investigated on a case by case basis depending upon the application. Consequently, the BMF framework developed in this paper represents an interesting starting point for a number of different extensions. 20
Cointegrating MiDaS Regressions and a MiDaS Test
"... This paper introduces cointegrating mixed data sampling (CoMiDaS) regressions, generalizingnonlinearMiDaSregressionsintheextantliterature. Underalinearmixedfrequency datagenerating process, MiDaS regressions provide a parsimoniously parameterized nonlinear alternative when the linear forecasting m ..."
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This paper introduces cointegrating mixed data sampling (CoMiDaS) regressions, generalizingnonlinearMiDaSregressionsintheextantliterature. Underalinearmixedfrequency datagenerating process, MiDaS regressions provide a parsimoniously parameterized nonlinear alternative when the linear forecasting model is overparameterized and may be infeasible. In spite of potential correlation of the error term both serially and with the regressors, I find that nonlinear least squares consistently estimates the minimum meansquared forecast error parameter vector. The exact asymptotic distribution of the difference may be nonstandard. I propose a novel testing strategy for nonlinear MiDaS and CoMiDaS regressions against a general but possibly infeasible linear alternative. An empirical application to nowcasting global real economic activity using monthly covariates illustrates the utility of the approach.
is given to the source. Facts and Challenges from the Great Recession for Forecasting and Macroeconomic Modeling
, 2013
"... our sole responsibility.�The views expressed herein are those of the authors and do not necessarily reflect the views of the National Bureau of Economic Research. At least one coauthor has disclosed a financial relationship of potential relevance for this research. Further information is available ..."
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our sole responsibility.�The views expressed herein are those of the authors and do not necessarily reflect the views of the National Bureau of Economic Research. At least one coauthor has disclosed a financial relationship of potential relevance for this research. Further information is available online at